Steiner Connectivity Augmentation and Splitting-off in Poly-logarithmic Maximum Flows

TL;DR

This paper presents an almost-linear time algorithm for Steiner connectivity augmentation and splitting-off problems, leveraging recent maximum flow algorithms to achieve efficiency in complex network connectivity tasks.

Contribution

It introduces a nearly-linear time algorithm for Steiner connectivity augmentation and splitting-off, extending previous work from global to Steiner connectivity with improved efficiency.

Findings

Achieves almost-linear running time for Steiner connectivity augmentation.

Extends the algorithm to Steiner splitting-off problem with similar efficiency.

Reduces the number of maximum flow calls from all-pairs to polylogarithmic in the splitting-off problem.

Abstract

We give an almost-linear time algorithm for the Steiner connectivity augmentation problem: given an undirected graph, find a smallest (or minimum weight) set of edges whose addition makes a given set of terminals $\tau$-connected (for any given $\tau > 0$). The running time of our algorithm is dominated by polylogarithmic calls to any maximum flow subroutine; using the recent almost-linear time maximum flow algorithm (Chen et al., FOCS 2022), we get an almost-linear running time for our algorithm as well. This is tight up to the polylogarithmic factor even for just two terminals. Prior to our work, an almost-linear (in fact, near-linear) running time was known only for the special case of global connectivity augmentation, i.e., when all vertices are terminals (Cen et al., STOC 2022). We also extend our algorithm to the closely related Steiner splitting-off problem, where the edges incident on a vertex have to be {\em split-off} while maintaining the (Steiner) connectivity of a given set of terminals. Prior to our work, a nearly-linear time algorithm was known only for the special case of global connectivity (Cen et al., STOC 2022). The only known generalization beyond global connectivity was to preserve all pairwise connectivities using a much slower algorithm that makes $n$ calls to an all-pairs maximum flow (or Gomory-Hu tree) subroutine (Lau and Yung, SICOMP 2013), as against polylog(n) calls to a (single-pair) maximum flow subroutine in this work.